Factors versus multiples
Factors divide a number: the factors of 12 are 1, 2, 3, 4, 6 and 12. Multiples are what you get by multiplying it: 12, 24, 36 and so on. The GCF looks for the biggest factor two numbers share; the LCM looks for the first multiple they share.
A worked pair
For 12 and 18 the GCF is 6 and the LCM is 36. The two are linked by GCF × LCM = 12 × 18 = 216. Another pair: 48 and 180 have a GCF of 12 and an LCM of 720. The GCF and LCM Calculator returns both at once.
Why they are useful
The GCF reduces fractions to lowest terms, and the LCM gives the smallest common denominator for adding them. They also solve timing and grouping problems, such as when two cycles line up again or how to split two groups into equal teams.
Other names
GCF is also called the highest common factor (HCF) or greatest common divisor (GCD). If two numbers share no factor other than 1, they are called coprime and their LCM is their product. For the full method, including Euclid’s algorithm, see how to find the GCF and LCM.